Kulfan Airfoil Visualizer

Class-Shape Transformation (CST) Method

The Kulfan parameterization, also known as Class-Shape Transformation (CST), provides a powerful method for representing airfoil geometries using a small number of parameters. This approach separates the "class" function (which defines the general shape characteristics) from the "shape" function (which defines the specific geometry).

Mathematical Formulation

The airfoil coordinates are defined by:

where:

• C(x) is the class function
• S(x) is the shape function
• t_te is the trailing edge thickness

Class Function

The class function defines the basic airfoil shape:

• N1 (Leading Edge): Controls leading edge radius. Lower values (0.3-0.5) create rounder leading edges, higher values (0.6-0.8) create sharper leading edges.
• N2 (Trailing Edge): Controls trailing edge shape. Values around 1.0 give typical airfoil trailing edges.

Shape Function

The shape function uses Bernstein polynomials:

where Bi,n(x) are Bernstein basis polynomials and Ai are the shape weights.

Thickness and Camber

The thickness and camber are controlled by scaling factors applied to the shape function:

• Thickness Factor: Scales the overall thickness of the airfoil
• Camber Factor: Adds camber by combining upper and lower surface modifications

Parameter Relationships

The sliders in this visualizer directly control:

• N1 Slider: Leading edge class parameter (0.3-0.8)
• N2 Slider: Trailing edge class parameter (0.5-1.5)
• TE Thickness: Trailing edge thickness as % of chord
• Leading Edge Weight: Additional leading edge shape modification
• Thickness Factor: Overall thickness scaling
• Camber Factor: Camber adjustment
• Bernstein Weights: Shape function coefficients for upper and lower surfaces

For more details, see: Kulfan, B.M. (2008). "Universal Parametric Geometry Representation Method." Journal of Aircraft, 45(1), 142-158.